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# Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

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We study asymptotical behavior of the probabilities of first-order properties for Erdős-Rényi random graphs G(n,p(n)) with p(n)=n-α, α ∈ (0,1). The following zero-one law was proved in 1988 by S. Shelah and J.H. Spencer [1]: if α is irrational then for any first-order property L either the random graph satisfies the property L asymptotically almost surely or it doesn't satisfy (in such cases the random graph is said to obey zero-one law. When α ∈ (0,1) is rational the zero-one law for these graphs doesn't hold.

Let k be a positive integer. Denote by Lk the class of the first-order properties of graphs defined by formulae with quantifier depth bounded by the number k (the sentences are of a finite length). Let us say that the random graph obeys zero-one k-law, if for any first-order property L ∈ Lk either the random graph satisfies the property L almost surely or it doesn't satisfy. Since 2010 we prove several zero-one \$k\$-laws for rational α from Ik=(0, 1/(k-2)] ∪ [1-1/(2k-1), 1). For some points from Ik we disprove the law. In particular, for α ∈ (0, 1/(k-2)) ∪ (1-1/2k-2, 1) zero-one k-law holds. If α ∈ {1/(k-2), 1-1/(2k-2)}, then zero-one law does not hold (in such cases we call the number α k-critical).

We also disprove the law for some α ∈ [2/(k-1), k/(k+1)]. From our results it follows that zero-one 3-law holds for any α ∈ (0,1). Therefore, there are no 3-critical points in (0,1). Zero-one 4-law holds when α ∈ (0,1/2) ∪ (13/14,1). Numbers 1/2 and 13/14 are 4-critical. Moreover, we know some rational 4-critical and not 4-critical numbers in [7/8,13/14). The number 2/3 is 4-critical. Recently we obtain new results concerning zero-one 4-laws for the neighborhood of the number 2/3.

References

[1] S. Shelah, J.H. Spencer, Zero-one laws for sparse random graphs, J. Amer. Math. Soc.

1: 97–115, 1988.

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### Maksim Zhukovskii – Zero-one k-laws for G(n,n−α)

1. 1. Zero-one k-laws and extended zero-one k-laws for random distance graphs Popova Svetlana vomonosov wosow tte niversity orkshop on ixtreml qrph heory wosowD tune TD PHIR 1/22
2. 2. he(nitionsX ird¥osE¡enyi rndom grph G(n, p) nd rndom grph G(Gn, p) he(nitionF ird¥osE¡enyi rndom grph G(n, p) is  rndom element with vlues in Ωn nd distriution Pn,p on FnD where Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn , Pn,p(G) = p|E| (1 − p)C2 n−|E| . 2/22
3. 3. he(nitionsX ird¥osE¡enyi rndom grph G(n, p) nd rndom grph G(Gn, p) he(nitionF ird¥osE¡enyi rndom grph G(n, p) is  rndom element with vlues in Ωn nd distriution Pn,p on FnD where Ωn = {(V = {1, ..., n}, E)}, Fn = 2Ωn , Pn,p(G) = p|E| (1 − p)C2 n−|E| . he(nitionF vet Gn e  sequene of grphs Gn = (Vn, En)F ndom grph G(Gn, p) is  rndom element with vlues in ΩGn nd distriution PGn,p on FGn D where ΩGn = {G = (V, E) : V = Vn, E ⊆ En}, FGn = 2ΩGn , PGn,p(G) = p|E| (1 − p)|En|−|E| . 2/22
4. 4. he(nitionsX (rstEorder properties nd zeroEone lw he(nitionF pirstEorder properties of grphs re de(ned y (rstEorder formuleD whih re uilt of predite symols ∼, = logil onnetivities ¬, ⇒, ⇔, ∨, ∧ vriles x, y, . . . qunti(ers ∀, ∃ 3/22
5. 5. he(nitionsX (rstEorder properties nd zeroEone lw he(nitionF pirstEorder properties of grphs re de(ned y (rstEorder formuleD whih re uilt of predite symols ∼, = logil onnetivities ¬, ⇒, ⇔, ∨, ∧ vriles x, y, . . . qunti(ers ∀, ∃ he(nitionF he rndom grph G(n, p) is sid to follow zeroEone lw if for ny (rstEorder property L either lim n→∞ Pn,p(L) = 0 or lim n→∞ Pn,p(L) = 1. 3/22
6. 6. he(nitionsX zeroEone kElw he(nitionF he rndom grph G(n, p) is sid to follow zeroEone kElw if for ny property L de(ned y  (rstEorder formul with qunti(er depth t most k either lim n→∞ Pn,p(L) = 0 or lim n→∞ Pn,p(L) = 1. 4/22
7. 7. eroEone lw for ird¥osE¡enyi rndom grph G(n, p) heorem@qleski et lFD IWTWY pginD IWUTA Let a function p = p(n) satisfy the property ∀β > 0 min(p, 1 − p)nβ → ∞ when n → ∞. Then the random graph G(n, p) follows the zero-one law. 5/22
8. 8. eroEone lw for ird¥osE¡enyi rndom grph G(n, p) heorem@qleski et lFD IWTWY pginD IWUTA Let a function p = p(n) satisfy the property ∀β > 0 min(p, 1 − p)nβ → ∞ when n → ∞. Then the random graph G(n, p) follows the zero-one law. heorem@helhD penerD IWVVA Let p(n) = n−β and β be an irrational number, 0 < β < 1. Then the random graph G(n, p) follows the zero-one law. 5/22
9. 9. ndom distne grph ndom distne grph G(Gdist n , p) Gdist n = (V dist n , Edist n ) a = a(n), c = c(n) V dist n = v = (v1 , . . . , vn ) : vi ∈ {0, 1}, n i=1 vi = a Edist n = {{u, v} ∈ V dist n × V dist n : (u, v) = c} 6/22
10. 10. eroEone lw for rndom distne grph vet  funtion p = p(n) stisfy the property ∀β > 0 min(p, 1 − p)|V dist n |β → ∞ when n → ∞. 7/22
11. 11. eroEone lw for rndom distne grph vet  funtion p = p(n) stisfy the property ∀β > 0 min(p, 1 − p)|V dist n |β → ∞ when n → ∞. heorem Let a(n) = αn, c(n) = α2n, α ∈ Q, 0 < α < 1. Then the random graph G(Gdist n , p) doesn't follow the zero-one law, but there exists a subsequence G(Gdist ni , p) following the zero-one law. 7/22
12. 12. uestions hen does  given susequene G(Gdist ni , p) follow zeroEone lwc 8/22
13. 13. uestions hen does  given susequene G(Gdist ni , p) follow zeroEone lwc hoes there exist  (rstEorder property L nd  susequene G(Gdist ni , p) suh tht lim i→∞ PGdist ni ,p(L) ∈ (0, 1) 8/22
14. 14. uestions hen does  given susequene G(Gdist ni , p) follow zeroEone lwc hoes there exist  (rstEorder property L nd  susequene G(Gdist ni , p) suh tht lim i→∞ PGdist ni ,p(L) ∈ (0, 1) ht limiting proilities PGdist ni ,p(L) n we getc 8/22
15. 15. ixtended zeroEone kElw he(nitionF he rndom grph G(Gn, p) is sid to follow extended zeroEone kElw if for every property L de(ned y  (rstEorder formul with qunti(er depth t most k ny prtil limit of the sequene PGn,p(L) equls either 0 or 1F 9/22
16. 16. ixtended zeroEone kElw he(nitionF he rndom grph G(Gn, p) is sid to follow extended zeroEone kElw if for every property L de(ned y  (rstEorder formul with qunti(er depth t most k ny prtil limit of the sequene PGn,p(L) equls either 0 or 1F qolF pind onditions on the sequene G(Gdist ni , p) under whih one of the following tkes pleX zeroEone kElw holds zeroEone kElw doesn9t holdD ut extended zeroEone kElw holds extended zeroEone kElw doesn9t hold 9/22
17. 17. ihrenfeuht gme EHR(G, H, k) EHR(G, H, k) qrphs G, HD numer of rounds k wo plyers poiler nd huplitor iEth roundX poiler hooses  vertex either from G or from H huplitor hooses  vertex of the other grph vet x1, . . . , xkD y1, . . . , yk e verties hosen from grphs G nd H respetivelyF huplitor wins if nd only if G|{x1,...,xk} ∼= H|{y1,...,yk}F 10/22
18. 18. ihrenfeuht gme EHR(G, H, k) EHR(G, H, k) qrphs G, HD numer of rounds k wo plyers poiler nd huplitor iEth roundX poiler hooses  vertex either from G or from H huplitor hooses  vertex of the other grph vet x1, . . . , xkD y1, . . . , yk e verties hosen from grphs G nd H respetivelyF huplitor wins if nd only if G|{x1,...,xk} ∼= H|{y1,...,yk}F heorem The random graph G(Gn, p) follows zero-one k-law if and only if P(Duplicator wins the game EHR(G(Gn, p), G(Gm, p), k)) → 1 as n, m → ∞. 10/22
19. 19. pull level extension property he(nitionF he grph G = (V, E) is sid to stisfy full level t extension property if for ny verties v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists  vertex v djent to v1, . . . , vl nd nonEdjent to u1, . . . , urF 11/22
20. 20. pull level extension property he(nitionF he grph G = (V, E) is sid to stisfy full level t extension property if for ny verties v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists  vertex v djent to v1, . . . , vl nd nonEdjent to u1, . . . , urF roposition Let G(Gn, p) satisfy full level (k − 1) extension property asymptotically almost surely. Then the random graph G(Gn, p) follows zero-one k-law. 11/22
21. 21. pull level extension property he(nitionF he grph G = (V, E) is sid to stisfy full level t extension property if for ny verties v1, . . . , vl, u1, . . . , ur (l + r ≤ t) there exists  vertex v djent to v1, . . . , vl nd nonEdjent to u1, . . . , urF roposition Let G(Gn, p) satisfy full level (k − 1) extension property asymptotically almost surely. Then the random graph G(Gn, p) follows zero-one k-law. gorollry Let G(Gn, p) satisfy full level t extension property a.a.s for every t ∈ N. Then the random graph G(Gn, p) follows the zero-one law. 11/22
22. 22. pull level extension property for rndom distne grph roposition Let a(n) = αn, α ∈ Q, 0 < α < 1. Then G(Gdist ni , p) satises full level t extension property a.a.s for every t ∈ N if and only if c = α2n and ∀m ∈ N m|ni for suciently large i. 12/22
23. 23. pull level extension property for rndom distne grph roposition Let a(n) = αn, α ∈ Q, 0 α 1. Then G(Gdist ni , p) satises full level t extension property a.a.s for every t ∈ N if and only if c = α2n and ∀m ∈ N m|ni for suciently large i. roposition Let a(n) = αn, c = α2n, α ∈ Q, 0 α 1, t ≤ 5. Then G(Gdist ni , p) satises full level t extension property a.a.s if and only if Dt|a(ni) − c(ni) for suciently large i, where D2 = 1, D3 = 2, D4 = 6, D5 = 60. 12/22
24. 24. eroEone kElws for rndom distne grph xottionF a = αn, c = α2n, α = s/q, (s, q) = 1. 13/22
25. 25. eroEone kElws for rndom distne grph xottionF a = αn, c = α2n, α = s/q, (s, q) = 1. heorem @zeroEone 4ElwA The random graph G(Gdist n , p) follows extended zero-one 4-law. The sequence G(Gdist ni , p) follows zero-one 4-law if and only if ∃i0 such that all the numbers a(ni) − c(ni) for i i0 have the same parity. 13/22
26. 26. eroEone kElws for rndom distne grph xottionF a = αn, c = α2n, α = s/q, (s, q) = 1. heorem @zeroEone 4ElwA The random graph G(Gdist n , p) follows extended zero-one 4-law. The sequence G(Gdist ni , p) follows zero-one 4-law if and only if ∃i0 such that all the numbers a(ni) − c(ni) for i i0 have the same parity. heorem @zeroEone 5ElwA Let a sequence {ni} be such that a(ni) − c(ni) are even for suciently large i. Then G(Gdist ni , p) follows extended zero-one 5-law, G(Gdist ni , p) follows zero-one 5-law if and only if ∃i0 such that either ∀i i0 3|a(ni) − c(ni) or ∀i i0 3 a(ni) − c(ni). 13/22
27. 27. eroEone kElws for rndom distne grph heorem @zeroEone 6ElwA Let q = 5 and a sequence {ni} be such that a(ni) − c(ni) are divisible by 12 for suciently large i. Then G(Gdist ni , p) follows extended zero-one 6-law, G(Gdist ni , p) follows zero-one 6-law if and only if ∃i0 such that either ∀i i0 5|a(ni) − c(ni) or ∀i i0 5 a(ni) − c(ni). 14/22
28. 28. hisproof of extended zeroEone lws for rndom distne grph ∀β 0 min(p, 1 − p)|V dist n |β → ∞ s n → ∞. (∗) heorem @disproof of extended zeroEone 6ElwA Let one of the following two cases take place: q = 5 and a sequence {ni} is such that a(ni) − c(ni) are not divisible by 5 for suciently large i, α = 1 2 and a sequence {ni} is such that a(ni) − c(ni) are not divisible by 4 for suciently large i. Then there exists a function p(n) satisfying (∗) such that G(Gdist ni , p) doesn't follow extended zero-one 6-law. 15/22
29. 29. hisproof of extended zeroEone lws for rndom distne grph heorem @disproof of extended zeroEone lwA Let q be even, α ∈ (1 4, 3 4) and a sequence {ni} be such that a(ni) − c(ni) are not divisible by 4 for suciently large i. Then there exists a function p(n) satisfying (∗) such that G(Gdist ni , p) doesn't follow extended zero-one law. 16/22
30. 30. peil sets of verties he(nitionF erties v1, . . . , vt of  grph G = (V, E) re sid to form  speil tEset if there doesn9t exist  vertex v ∈ V djent to ll of the verties v1, . . . , vtF 17/22
31. 31. peil sets of verties he(nitionF erties v1, . . . , vt of  grph G = (V, E) re sid to form  speil tEset if there doesn9t exist  vertex v ∈ V djent to ll of the verties v1, . . . , vtF vet Rt e  property of spnning sugrphs of GnX for ny verties v1, . . . , vt not forming  speil tEset in Gn nd for ny suset U ⊆ {v1, . . . , vt} there exists  vertex v djent to ll verties from U nd nonEdjent to ll verties from {v1, . . . , vt} UF 17/22
32. 32. peil sets of verties he(nitionF erties v1, . . . , vt of  grph G = (V, E) re sid to form  speil tEset if there doesn9t exist  vertex v ∈ V djent to ll of the verties v1, . . . , vtF vet Rt e  property of spnning sugrphs of GnX for ny verties v1, . . . , vt not forming  speil tEset in Gn nd for ny suset U ⊆ {v1, . . . , vt} there exists  vertex v djent to ll verties from U nd nonEdjent to ll verties from {v1, . . . , vt} UF roposition For every t ∈ N the random graph G(Gdist n , p) satisfyes Rt a.a.s. 17/22
33. 33. roof of zeroEone kElwsX speil sets of verties without edges uppose @IA Gn = (Vn, En) doesn9t hve speil (t − 1)Esets @PA G(Gn, p) stisfyes Rt FFsF 18/22
34. 34. roof of zeroEone kElwsX speil sets of verties without edges uppose @IA Gn = (Vn, En) doesn9t hve speil (t − 1)Esets @PA G(Gn, p) stisfyes Rt FFsF roposition Let a sequence Gn = (Vn, En) satisfy (1), (2) and the following conditions: Gn has special t-sets, for every special t-set any two of its vertices are non-adjacent. Then the random graph G(Gn, p) follows zero-one (t + 1)-law. 18/22
35. 35. roof of zeroEone kElwsX speil sets of verties with edges roposition Suppose Gn = (Vn, En) satises (1), (2) and for any vertices v1, . . . , vi where i t one of the following holds: for any vertex vi+1 such that v1, . . . , vi+1 can be extended to a special t-set there exist Ω(|Vn|β) dierent vertices each of which can be mapped onto vi+1 by an automorphism of Gn xing v1, . . . , vi (where β is a positive constant), |{(vi+1, . . . , vt) : {v1, . . . , vt} is a special t-set}| = O(1). Then the random graph G(Gn, p) follows extended zero-one (t + 1)-law. 19/22
36. 36. hisproof of extended zeroEone kElws vet L e  property of sugrphs G ⊆ GnX for ny (v1, . . . , vi) tht n e extended to  speil tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to  speil tEset with edges in GF 20/22
37. 37. hisproof of extended zeroEone kElws vet L e  property of sugrphs G ⊆ GnX for ny (v1, . . . , vi) tht n e extended to  speil tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to  speil tEset with edges in GF vet K(v1, . . . , vi) e the numer of (vi+1, . . . , vt) extending (v1, . . . , vi) to  speil tEset with edges in GnF 20/22
38. 38. hisproof of extended zeroEone kElws vet L e  property of sugrphs G ⊆ GnX for ny (v1, . . . , vi) tht n e extended to  speil tEset with edges in Gn there exist vi+1, . . . , vt extending (v1, . . . , vi) to  speil tEset with edges in GF vet K(v1, . . . , vi) e the numer of (vi+1, . . . , vt) extending (v1, . . . , vi) to  speil tEset with edges in GnF sf there exists (v1, . . . , vi) with K(v1, . . . , vi) → ∞, K(v1, . . . , vi) = |Vn|o(1) , then PGn,p(L) n pproh ny numer from (0, 1)F 20/22
39. 39. hisproof of extended zeroEone kElws eple L y  (rstEorder property LX L = ∀v1 . . . ∀vi ∃vi+1 . . . ∃vt Q(v1, . . . , vt), where Q pproximtely sys tht either (v1, . . . , vi) n9t e extended to  speil tEset with edges in Gn or (v1, . . . , vt) forms  speil tEset with edges in G(Gn, p)F 21/22
40. 40. hnks Thank you for your attention! 22/22